Transfer function inversion methodology

In order to obtain a well-defined inverse transformation, the transformation itself must be well-defined and it must characterise the system without excessive complexity. Inversion is based on differentiation and is, therefore, numerically poorly defined by definition. In this study, a novel solution to estimating Discharge Generating Rainfall has been proposed utilising the inverse of a continuous-time transfer function and regularisation, termed the RegDer method (Kretzschmar et al., 2014). It is compared to the direct inverse of a transfer function (Andrews et al., 2010) termed InvTF in chapter 4.

The general inversion of a linear system, as described in Section 1.3, is shown in Figure 2-4. If G ([: O −> M) is a true representation of the system and G-1([AB_{: O}∗ _{−> M}∗₎ is the true dynamic inverse then the overall system input, U, is the same as the output, Y* (Buchholz and Grünhagen, 2004)

Figure 2-4 - A schematic representation of a general identity system assuming a perfect model and a perfect inverse. In the ideal case, the system input U is identical to the system output Y* (adapted from Buchholz and Grünhagen, 2004)

The general system, G, can be represented a transfer function of the form:

[ = 6(%)_<(%) (Equation 2-17)

with A and B polynomials of orders n and m respectively. Transfer functions are linear operators so once the time delay and parameters have been identified, the model can be rewritten as:

G

U

G-1

Y*

29 [AB₌ <(%)

6(%) (Equation 2-18)

from which a series of linearised rainfalls may be inferred. As shown in Figure 2-4, in an ideal situation, the series of inferred rainfalls would exactly equal the rainfall inputs to the system. In the real world, this is unlikely and how well Y* matches U is dependent on the rainfall regime, the catchment dynamics, the quality of the data and how well the model represents the physical processes as well as the efficiency of the inversion.

A ‘proper’ transfer function depends on the relative orders of the numerator (order m) and denominator (order n). To be proper, n >= m is required. If this is not the case, the TF model will not be realisable due to the fact that perfect derivatives do not exist (as involving knowledge of the future), and thus will be rejected by DBM methodology. An improper TF can be seen to be responding to future events now, which is clearly impossible (Dokuchaev, 2016). This problem arises with the inverse TF models – a common situation with many systems being ‘strictly proper’ where n > m resulting in an inverse where the opposite is true, which therefore is unrealisable.

Two approaches to resolving this issue are taken in this study. 2.5.1. Regularised derivative estimate approach

The Regularised Derivative method was developed from an idea first mooted by Jakeman and Young (1984) in combination with developments in the identification of CT-TF models (for example, Young and Garnier, 2006). The transfer function is inverted as in Equation 2-18 but is then split into a ‘proper’ realisable part and the unrealisable part which will require the use of derivatives. The realisable part takes Y, the original system output, as its input whilst the part requiring derivatives, uses regularised estimates of the derivatives as input.

Regularisation is a mathematical technique that introduces extra information allowing an ill-posed problem to be solved numerically. The additional information in this case takes the form of imposing a loss of temporal resolution (increasing the smoothness of the solution), thus effectively limiting the number of estimated values or parameters, and so simplifying the model. This process is sometimes interpreted as imposition of

Chapter 2 Background to methods

30 Occam’s Razor on the solution (the law of parsimony which states that the simplest answer is often correct). Regularisation is necessary here because minimising the objective function (residual sum of squares) leads to exaggeration of high frequency components of the estimated signal, particularly for catchments with large storage and slow and multiple time constants. The tuning parameter introduced is the NVR (equation 4-9) which is reciprocally related to the smoothness of the estimate. It is applied only to the higher derivative estimates (c.f. equation 4-4) allowing the amount of smoothing and therefore loss of resolution to be tuned to give the best fit to the observed rainfall.

The derivative estimates are obtained using a regularisation technique using higher order Integrated Random Walk models in a stochastic state-space framework, a technique available in the Captain Toolbox for Matlab (Taylor et al., 2007). For detailed explanation, section 4.3. The following example shows a CT-TF model linking linearised rainfall and streamflow:

( = <(%)_6(%) !P = $S%U $:

%]_{U V}

:%UV] !P (Equation 2-19)

where the order of the numerator m=1 and the order of denominator n=2. This is a proper TF where n>m so when the inverse is written as:

!P = 6(%)

<(%)( =

%]U V:% U V]

$S%U $: Q (Equation 2-20)

where the order of the numerator m=2 and the order of denominator n=1. This is an improper TF because n<m and it involves a pure derivative of Q. It can be transformed to: !P∗₌ %]U V:% U V] $S%U $: ( = % $S%U $: N( + V:% U V] $S%U $: Q (Equation 2-21) where the TFs are proper but involves the derivative of Q, sQ, which is estimated using the regularised estimate of the derivative of Q obtained from an Integrated Random Walk model of Q (IRWSM in the Captain toolbox) (Jakeman and Young, 1984; Young et al., 1999). The inverse transform can be rewritten using the regularised derivative estimate approximation (sQ)*:

31 !P∗₌ %

$S%U $: N(

∗₊ V:% U V]

$S%U $: Q (Equation 2-22)

which is realisable and straightforward to implement.

2.5.2. The alternative fast compensating mode approach

Equation 2-18 can be rewritten in a realisable (or ‘proper’, that is not involving direct derivatives) form (Zadeh and Desoer, 1963) given by:

[AB₌ <(%) 6(%)

$(%)

V(%) (Equation 2-23)

where _V(%)$(%)is a compensating transfer function which makes the overall inverse realisable with no pure differentiation. The order of the denominator a(s) is chosen to be of an order such that the overall denominator is of higher order than the numerator. The compensating TF has a SSG of 1 and the roots of the numerator (poles) are chosen to be fast, that is, well above the upper range of the original model spectrum so that the inverse dynamics are not affected. The Direct Inverse approach taken by Andrews et al., (2010) was used as comparison with the novel regularised derivative method in

chapter 4.